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A precise prediction of maximum scour depth around bridge foundations under ice covered condition is crucial for their safe design because underestimation may result in bridge failure and over-estimation will lead to unnecessary construction costs. Compared to pier scour depth predictions within an open channel, few studies have attempted to predict the extent of pier scour depth under ice-covered condition. The present work examines scour under ice by using a series of clear-water flume experiments employing two adjacent circular bridge piers in a uniform bed were exposed to open channel and both rough and smooth ice covered channels. The measured scour depths were compared to three commonly used bridge scour equations including Gao’s simplified equation, the HEC-18/Jones equation, and the Froehlich Design Equation. The present study has several advantages as it adds to the understanding of the physics of bridge pier scour under ice cover flow condition, it checks the validity and reliability of commonly used bridge pier equations, and it reveals whether they are valid for the case of scour under ice-covered flow conditions. In addition, it explains how accurately an equation developed for scour under open channel flow can predict scour around bridge piers under ice-covered flow condition.

Scouring can be defined as the process by which the particles of soil or rock around an abutment or pier of a bridge get eroded and removed to a certain depth (called the scour depth) [

Most of the equations for the prediction of bridge pier scouring express the final scour depth as a function of the flow characteristics (mean flow velocity at the approach section, water depth), flow properties (density and viscosity of the fluid), stream bed material properties (mean particle diameter, density) and bridge geometry (shape and dimension of the pier, angle of attack of the flow). In this paper, scour around circular bridge piers will be experimentally examined and subsequently the validity and reliability of three of the more commonly used and cited scour equations developed specifically for open channel flow condition will be investigated to see how accurately they predict scouring around bridge piers under ice-covered flow conditions.

The most commonly used pier scour equation in the United States is the Colorado State University (CSU) equation proposed by Richardson and Davis [

y s / y 0 = 2 K 1 K 2 K 3 K 4 ( b / y 0 ) 0.65 F r 0.43 (1)

where y_{s} = scour depth; y_{0} = the approach flow depth; y_{s}/y_{0} is a dimensionless expression of the relative scour depth with respect to flow depth; K_{1} = correction factor for pier nose shape which is unity for circular cylinder; K_{2} = correction factor for angle of attack flow which is unity for 90^{0}; K_{3} = correction factor for bed condition which is 1.1 for clear water scour; b = nominal pier width; and Fr = approach flow Froude number. K_{4} is a correction factor to account for armoring of the scour hole:

K 4 = [ 1 − 0.89 ( 1 − V R ) 2 ] 0.5 (2)

where V_{R} is the velocity ratio and is dimensionless:

V R = [ V 0 − V i 50 V c 90 − V i 50 ] (2a)

where V_{o} is the approach velocity directly upstream from the pier and V_{i50} is the approach velocity, in feet per second, required to initiate scour at the pier for the particle size D_{50}. V_{i}_{50} is calculated as follows:

V i 50 = 0.645 [ D 50 b ] 0.053 V c 50 (2b)

where D_{50} is the particle size for which 50 percent of the bed material is finer, in units of feet and V_{c}_{50} is the critical velocity, in feet per second, for incipient motion of the particle size D_{50}. V_{c}_{50} is defined as follows:

V c 50 = 11.21 y 0 1 / 6 D 50 1 / 3 (2c)

while D_{90} is the particle size for which 90 percent of the bed material is finer, in units of feet, and V_{c}_{90} is the critical velocity, in feet per second, for the incipient motion of the particle size as given by:

V c 90 = 11.21 y 0 1 / 6 D 90 1 / 3 (2d)

Gao’s simplified pier scour equation is based on laboratory and field data from China [

y s = 1.141 K s b 0.6 y 0 0.15 D m − 0.07 ( V 0 − V i c V c − V i c ) (3)

where y_{s} is the depth of pier scour below the ambient bed, in feet; K_{s} is the simplified pier shape coefficient which is 1.0 for cylinders; b is the width of bridge pier, in feet; y_{0} is the depth of flow directly upstream from the pier, in feet; D_{m} is the mean particle size of the bed material, in feet (for this study D_{50} was used as D_{m}); V_{o} is the approach velocity directly upstream from the pier, in feet per second; and V_{c} is the critical (incipient motion) velocity, in feet per second, for the D_{m}-sized particle. V_{ic} is the approach velocity, in feet per second, corresponding to critical velocity at the pier. V_{ic} can be calculated using the following equation:

V i c = 0.645 ( D m b ) 0.053 V c (4)

If the density of water is assumed to be 62.4 pounds per cubic foot and the bed material is assumed to have a specific gravity of 2.65, the equation for V_{c} can be expressed as:

V c = 3.28 ( y 0 D m ) 0.14 ( 8.85 D m + 6.05 E − 7 [ 10 + 0.3048 y 0 ( 0.3048 D m ) 0.72 ] ) 0.5 (5)

The Froehlich design equation is included as a pier-scour calculation option within the computer model HEC-RAS, Version 3.1 [

y s = 0.32 b ϕ F r 1 0.2 ( b e b ) 0.62 ( y 0 b ) 0.46 ( b D 50 ) 0.08 + b (6)

where ϕ is a dimensionless coefficient based on the shape of the pier nose, and is 1.0 for round-nosed piers; Fr_{1} is the Froude Number directly upstream from the pier; b_{e} is the width of the bridge pier projected normal to the approach flow, in feet; b is the width of the bridge pier, in feet; D_{50} is the particle size for which 50 percent of the bed material is finer, in feet and y_{0} is the depth of flow directly upstream from the pier, in feet.

Experiments were carried out at the Quesnel River Research Centre, Likely, BC, Canada in a large-scale flume. The flume measures 40 m long, 2 m wide and 1.3m deep. The longitudinal slope of the flume bottom was 0.2 percent. A holding tank with a volume of nearly 90 m^{3} was located at the upstream end of the flume and provided a constant head in the experimental zone. Two valves were connected to the holding tank to allow for control of the flow velocity. At the end of the holding tank and upstream of the main flume, water overflowed from a rectangular weir into the flume. Since the flow of water was turbulent while entering the flume, a flow diffuser was placed downstream of the rectangular weir to dissipate the turbulence in the flow of water. Two sand boxes with the depth of 0.30 m were filled with a uniform sediment having a median particle size (D_{50}) of 0.47 mm. The first sand box was 5.6 m in length and the second sand box was 5.8 m in length. The distance between the sand boxes was 10.2 m. Four different pairs of bridge piers with diameter of 6 cm, 9 cm, 11 cm and 17 cm were used (

to manually verify water depth. The scour hole velocity field was measured using a 10-Mhz Acoustic Droppler Velocimeter (ADV) by SonTek. In order to simulate ice cover, 13 panels of Styrofoam with dimensions of 1.2 m × 2.4 m (4 × 8 foot) were used to cover nearly the entire surface of flume. Styrofoam density was 0.026 gr/cm^{3} and the Styrofoam was floated on the surface in the flume during the experimental runs. In the present study, two types of model ice cover were used, namely smooth cover and rough cover. The smooth ice cover was the surface of the original Styrofoam panels while the rough ice cover was made by attaching small Styrofoam cubes to the bottom of the smooth cover. The dimensions of Styrofoam cubes were 2.5 cm × 2.5 cm × 2.5 cm and were spaced 3.5 cm apart for the Styrofoam covering panels which were placed above the sand boxes and were spaced 5.5 m apart for the rest of the Styrofoam covering panels outside of the sand boxes. 36 experiments were conducted under open channel, smooth and rough ice conditions.

Maximum scour depths calculated by the above three pier-scour equations were compared to 36 sets of experimental data. In this section, comparisons between the results from each equation and the experimental results are discussed along with an Error analysis including RMSE (Root Mean Squared Error), Index of Agreement (I_{a}) and MAE (Mean Absolute Error). The Absolute Error (MAE); Root Mean Square Error (RMSE) and Index of Agreement (I_{a}) are mathematically described by the following equations:

MAE = ∑ i = 1 n | e i | n (7)

RMSE = ∑ i = 1 n ( y i − x i ) 2 n (8)

I a = 1 − ∑ i = 1 n ( y i − x i ) 2 ∑ i = 1 n ( | x i − x ¯ | + | y i − x ¯ | ) 2 (9)

where x_{i} is scour depth obtained from experiments and y_{i} is the corresponding predicted scour depths; x ¯ is the mean experimental scour depth and n is number of records. Smaller values of MAE and RMSE indicate a more successful prediction. The Index of Agreement (I_{a}) is a standardized measure of the degree of model prediction error and varies between 0 and 1. A value of 1 indicates a perfect match, while 0 indicates no agreement [

Equation | ||||||||
---|---|---|---|---|---|---|---|---|

Cover | Pier Identification | Measured Scour (ft.) | Gao’s simplified | Froehlich Design | Hec-18/Jones | |||

Calculated scour (ft.) | Residual (ft.) | Calculated scour (ft.) | Residual (ft.) | Calculated scour (ft.) | Residual (ft.) | |||

Smooth | Right | 0.10 | 0.06 | −0.03 | 0.28 | 0.19 | 0.24 | 0.15 |

Smooth | Left | 0.11 | 0.10 | −0.01 | 0.26 | 0.15 | 0.25 | 0.13 |

Smooth | Right | 0.14 | 0.10 | −0.04 | 0.29 | 0.15 | 0.28 | 0.14 |

Smooth | Left | 0.10 | 0.11 | 0.01 | 0.40 | 0.30 | 0.33 | 0.23 |

Smooth | Right | 0.22 | 0.19 | −0.03 | 0.38 | 0.15 | 0.36 | 0.14 |

Smooth | Left | 0.21 | 0.16 | −0.05 | 0.41 | 0.20 | 0.39 | 0.18 |

Smooth | Left | 0.14 | 0.08 | −0.06 | 0.47 | 0.33 | 0.33 | 0.20 |

Smooth | Right | 0.26 | 0.19 | −0.07 | 0.45 | 0.18 | 0.39 | 0.13 |

Smooth | Left | 0.26 | 0.18 | −0.08 | 0.49 | 0.23 | 0.43 | 0.17 |

Smooth | Left | 0.10 | 0.08 | −0.02 | 0.71 | 0.61 | 0.42 | 0.33 |

Smooth | Left | 0.16 | 0.17 | 0.01 | 0.67 | 0.51 | 0.46 | 0.30 |

Smooth | Left | 0.16 | 0.20 | 0.04 | 0.72 | 0.56 | 0.55 | 0.39 |

Rough | Left | 0.15 | 0.09 | −0.06 | 0.28 | 0.13 | 0.27 | 0.11 |

Rough | Right | 0.19 | 0.10 | −0.09 | 0.26 | 0.07 | 0.25 | 0.06 |

Rough | Left | 0.18 | 0.14 | −0.04 | 0.30 | 0.11 | 0.31 | 0.13 |

Rough | Right | 0.31 | 0.10 | −0.21 | 0.40 | 0.10 | 0.32 | 0.02 |

Rough | Right | 0.31 | 0.13 | −0.18 | 0.37 | 0.07 | 0.32 | 0.01 |

Rough | Left | 0.24 | 0.15 | −0.09 | 0.41 | 0.17 | 0.38 | 0.14 |

Rough | Left | 0.22 | 0.11 | −0.11 | 0.47 | 0.25 | 0.36 | 0.14 |

Rough | Right | 0.26 | 0.19 | −0.08 | 0.45 | 0.18 | 0.38 | 0.12 |

Rough | Right | 0.28 | 0.17 | −0.11 | 0.49 | 0.21 | 0.42 | 0.15 |

Rough | Left | 0.16 | 0.12 | −0.05 | 0.70 | 0.54 | 0.46 | 0.30 |

Rough | Left | 0.19 | 0.18 | −0.01 | 0.67 | 0.48 | 0.47 | 0.28 |

Rough | Right | 0.20 | 0.18 | −0.02 | 0.73 | 0.53 | 0.55 | 0.35 |

RMSE (%) | I_{a} | MAE (%) | |||||||
---|---|---|---|---|---|---|---|---|---|

SCE | HJE | FDE | SCE | HJE | FDE | SCE | HJE | FDE | |

Open | 3.58 | 24.88 | 35.50 | 0.90 | 0.48 | 0.34 | 6.8 | 23 | 38 |

Smooth | 4.38 | 22.33 | 33.91 | 0.84 | 0.56 | 0.41 | 14.1 | 26 | 25 |

Rough | 10.33 | 18.21 | 29.19 | 0.46 | 0.78 | 0.64 | 15.7 | 45 | 39 |

a comparison of the predicted maximum scour depth from the three equations to maximum measured scour depth under smooth and rough ice-covered flow condition. It should be mentioned the residual values are defined as predicted scour depth minus measured scour depth. A negative residual value indicates over-estimated scour depth. From the data provided, it is possible to determine which equation is most useful under various conditions. Evaluation of these equations are important to bridge design especially for the case of flow under ice-covered condition.

The scatterplots in

Overall, the most reliable and accurate equation which has predicted the pier scour depths under open channel and ice-covered flow to a very good extent was Gao’s simplified equation. Pier-scour depths calculated using the Gao’s simplified equation were smaller than measured scour depths for 6 of the 12 measurements for open channel flow and for 9 of the 12 measurements for smooth ice cover. However, it completely underestimated Pier-scour depths for the rough ice cover flow (

In terms of smooth ice-covered flow, the averages of pier-scour depths for open channel flow calculated from Froehlich equation was 0.46 ft, from the

HEC-18/Jones was 0.37 ft, and from the Gao’s simplified equation was 0.13 ft. The actual average measured pier-scour depth was 0.16 ft again indicating Gao’s simplified equation provided the best agreement with the measured value and was more successful in predicting the scour depth. Both the Froehlich and HEC-18/Jones equations resulted in the significant over-estimation of the scour depth on average. The highest index of agreement value (0.84) and the lowest RMSE and MAE values were obtained with Gao’s simplified equation under the smooth ice-covered flow condition.

In terms of rough ice-covered flow, the averages of pier-scour depths for open channel flow calculated from Froehlich equation was 0.46 ft, from the HEC-18/Jones was 0.37 ft, and from the Gao’s simplified equation was 0.14 ft. In this case, the rough ice surface increased the average measured pier-scour depths to 0.22 ft which is higher than the average scour depth calculated using Gao’s simplified equation. However, the results from Gao’s equation were closer to the experimental results than either the Froehlich or the HEC-18/Jones equation might suggest it was more successful in calculating scour depth. Further, the highest index of agreement value (0.78) and the lowest RMSE and MAE values are for the Gao’s simplified equation under rough ice-covered flow condition. Mathematically, Gao’s simplified equation is the most successful of the three equations at calculating scour depth but it does underestimate the extent of scour. In practical terms, this could lead to a false sense of security in practical situations as increased scour depth could lead to premature failure of a pier.

Three pier-scour equations, namely the Froehlich design equation, the HEC-18/ Jones equation and Gao’s simplified equation, were evaluated against the data from 36 flume experimental runs obtained for open channel, rough and smooth ice-covered flow under uniform bed sediment type. The most important result that can be obtained from the comparison of these three equations is that under nearly the same flow depth and approach flow velocity but different flow cover, the average calculated values from all the three equations stayed nearly constant. On the other hand, in terms of ice cover, the rougher the ice surface, the more turbulent the flow and the deeper the scour depth generated. Although Gao’s simplified equation was reasonably successful in prediction of pier scour depth for both the open-channel and smooth-ice conditions, it underestimated the pier-scour depth under rough ice-covered flow conditions. Therefore, it can be concluded that none of the equations adequately model scour depth under rough ice conditions and the equations are in need of another term to make them more suitable to be used for the ice-covered flow conditions.

This work is partially supported by UNBC Research Project Award. Physical experiments were conducted at Quesnel River Research Center (QRRC), Likely, BC. We gratefully acknowledge the help provided by QRRC staff.

Namaee, M.R., Li, Y.Q., Sui, J.Y. and Whitcombe, T. (2018) Comparison of Three Commonly Used Equations for Calculating Local Scour Depth around Bridge Pier under Ice Covered Flow Condition. World Journal of Engineering and Technology, 6, 50-62. https://doi.org/10.4236/wjet.2018.62B006